Arm chair remodeling - fun ! especially when you don't have to do the work ;D
Sounds like OSB (Oriented Strand Board) under the linoleum.
Room looks tiny - say 8' X ? 10 - 15'
First floor ? Whats under the floor ? A plaster ceiling ? or craw space, basement ?
Is the particle board damp, flaky. moldy, ?
If it's first floor and the room is as tiny as it looks I'm thinking rip it all out down to joists and sister, then some fresh 3/4" Plytanium sub floor and another 1/2" of ply on top of that (if needed) to finish level transition with other room then 1/4" hardibacker or wonderboard for tile.
If its a second floor or up and you have a ceiling below then I'm thinking the level goop over a decent surface. So as not to disturb the ceiling below, and whatever is in there like pipes, wiring and all those suprises that jump up and bite ya. All that demolition could shake loose an old plaster ceiling.
By now Per's crew is working on grouting and clean up